The 1997 PCMI Undergraduate Program
The beginning course
Since symplectic geometry in the modern sense grew out of
Lagrangian mechanics and solution methods for ordinary differential
equations, the theme of the program for the beginning undergraduate
group will be ordinary differential equations, particularly the
study of how conservation laws and solution methods based on them
interact with the qualitative and numerical study of classical
mechanical problems and their modifications. We'll be looking for
students who have had linear algebra, vector calculus, and a first
course in differential equations. (This is a common background for
sophomores and advanced first-year students.) We'll study how
first integrals of equations can be used to study the accuracy
of numerical integration schemes and how they can be used to taylor
iteration schemes to improve accuracy in particular cases. We'll
explore the geometry of equations that arise from variational
principles and see, with a minimum of machinery, how symmetries in
the system give rise to conservation laws. The students will also
learn how to use mathematical tools, such as Maple, MathCad, and
3-D Filmstrip to study differential equations and their applications
in geometry and mechanics.
The advanced course
The advanced students will follow a course designed to
lead them to the fundamentals of symplectic geometry in the modern
sense. We will be looking for students who have had the basic
courses in vector calculus and differential equations but who
have also had an introductory course to differential geometry
(at least at the level of curves and surfaces). The main theme
will be the interaction of symmetry groups and conservation laws,
culminating in the formulation of symplectic geometry as a unifying
theory for several distinct classical subjects. I intend to base
the lecture part of the course on parts of my "Lectures on
Symplectic Geometry and Lie Groups" (which was successfully offered
to the advanced undergraduates and beginning graduate students
at the very first Summer Institute in 1990). However, I will
also bring in some relations with stability and integrability of
classical ODE with KAM theory, solitons, fluid dynamics as possible
side topics. The students will experiment with computer modeling
of sympectic flows, the Liouville measure, geodesics on surfaces,
and other models as time permits.
The Faculy Enhancement component
The main benefits for participants in the Faculty Enhancement
program will be as follows: In the past, ordinary differential
equations has generally been taught in two very different modes: The
more classical mode is that of learning explicit solution techniques
for special classes of equations and their applications. In more
recent years, the 'modern' approach tends to concentrate on numerical
solutions and qualitative theory. In this summer program, we will
be trying to develop an intellectually satisfying synthesis of these
approaches, one that we believe will offer useful insights into how
the standard ODE courses can be improved. For example, since many
physically interesting ODEs arise through variational analysis, the
underlying symmetries (via Noether's theorem) can be used to construct
first integrals of the equations. These first integrals then can be
used to reduce the complexity or numerical sensitivity of computer-based
integration schemes. We'll be discussing how these ideas can be
developed and used without resorting to ideas that are often only
encountered very late (if at all) in an undergraduate's career and how
they can be used to motivate geometric and qualitative thinking
in the study of ordinary differential equations.
Robert L. Bryant <bryant@math.duke.edu>
email address: bryant@math.duke.edu
office 'phone: (919)-660-2805
office number: 128A Physics
Back to Robert Bryant's home page.
Updated: 29 September 1996